On an Integrable Variant Blaszak–Szum Lattice Equation

ABSTRACT We derive a novel variant of the Blaszak–Szum lattice equation by introducing a new class of trigonometric‐type bilinear Hirota operators. Gram‐type determinant solutions for the variant Blaszak–Szum lattice equation are obtained using the determinant technique and KP‐Toda hierarchy reduction approach, respectively. One‐soliton and two‐soliton solutions are constructed, with a detailed analysis of the asymptotic behaviors of the two‐soliton solution. Through parameter variation, three types of breather solutions are obtained, including the Akhmediev breather, Kuznetsov–Ma breather, and a general breather propagating along arbitrary oblique trajectories. A Bäcklund transformation for the variant Blaszak–Szum lattice equation is established, by virtue of which multi‐lump solutions of the equation are further constructed. Rational solutions are derived by introducing two differential operators applied to the determinant elements; in particular, the lump solutions derived via these differential operators can be formulated in terms of Schur polynomials. We investigate the large‐time patterns of higher order lump solutions and demonstrate that the distribution of the lump centers is analytically determined by the root structures of the Yablonskii–Vorob'ev polynomials. Finally, numerical three‐periodic wave solutions for both the Blaszak–Szum lattice equation and its variant are computed using the Gauss–Newton method.